When will she pass me for the first time?

[Guest post by Dr. Chase]

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

Example:  For m = 4, and n = 3, she will catch up when I have finished 3 laps.  Reason:  When I have finished 1 lap, she finished 1 1/3 laps, so she is 1/3 of the track ahead of me.  (But hasn’t passed me yet.)  When I have finished 2 laps, she has finished 2 2/3 laps around the track, still ahead of me.  When I finish 3 laps, she has finished 3 3/3 laps, which is to say 4 laps.  So we are together for the first time since starting.  If m = 2, n = 1, she will catch up in just 1 lap.  If m = 7, n = 6, she will catch up in 6 laps.  Will she always catch up in n laps?  In how many laps will she catch up for arbitrary m and n?

Chinese bridge inspired by Möbius band


[Guest post by Dr. Chase]

Is THIS bridge pictured above in the shape of a Möbius band or merely “associated” with a Möbius band as the article suggests?  If it is a Möbius band, where is the half-twist?  Do you think that the bridge is beautiful?  The architects have proposed that such a bridge be built in China.

Can you imagine a Möbius band being used for a road?  There was “A subway named Möbius,” to quote the title of a light-hearted 1950 short story by A. J. Deutsch.  It was published in the wonderful 1958 book Fantastia Mathematica.

The bridge above is only a concept.  Other one-sided surfaces have inspired architectural designs that have actually been built.  Here’s a house made in the shape of a Klein bottle.

A bit of mathematical humor.  One person comments on the Klein bottle that he likes the house’s orientation.  Well, if it were a true Klein bottle, it wouldn’t be orientable at all!

The Mathematics of Juggling and more from George Hart

[Dr. Chase guest blogging again]

The Mathematics of Juggling You’re probably familiar with Vi Hart’s math videos. Less well-known are her father’s math videos. Although I was aware of his mathematical sculpture, I was not aware until today that since August 2012, he has been producing a mathematical video series called mathematical impressions for the Simons Foundation. The 10th in the series is The Mathematics of Juggling. Check it out!

Flat Donuts

[Guest blogger Dr. Gene Chase]

You know about flat donuts if you played a computer game in which when you go off the screen at the left, you return coming in at the right, and similarly when you go off the screen at the top, you return coming in at the bottom.

Mathematicians call the rectangle of your computer screen a flat torus. The word “torus” reminds us that we are only thinking of the surface of the donut.

What does that computer screen have to do with a torus? Stretch the screen around a torus as this picture begins to show.

Flat Donut Being Wrapped around Curved Donut

Since the left and right sides of the screen are “the same” and the top and bottom sides of the screen are “the same,” the screen seamlessly takes the shape of the donut. Mathematicians say that the flat donut and the donut have the same topology, because we bent one into the other without cutting or pasting. (I remind you that opposite edges of the flat donut were already pasted by regarding them as “the same” before we began to bend it.)

Although a flat torus and a torus have the same topology, they do not have the same geometry. Geometry is about measuring space. In particular, on a flat torus, the shortest distance between two points is always a straight line. But on a torus, the shortest distance between two points staying in the torus is never a straight line.

Is it possible to paste the opposite edges of the flat torus together in such a way that the resulting thing in 3D has the same geometry? That is, such that straight lines are still straight?

It is easy to take one step, to create a cylinder, by pasting together just one pair of opposite edges. Notice that no stretching is involved at all.

Cylinder with square grid

How about pasting both pairs of edges? My intuition says that such a thing is impossible.

My intuition is wrong. In 1954, John Nash — yes, that John Nash of A Beautiful Mind — proved that it is possible, but without saying how. (He gave a so-called “existence proof.”)

But only 11 months ago did we have a picture of what the resulting flat torus would look like. Here’s a news article with a computer-generated picture to illustrate it.

Flat Torus Embedded in 3D Isometrically

Geometry Is Beautiful

Header of Dimensions

Find nine 14-minute videos on geometry here. They’re breathtakingly beautiful!.

  • 1 Mapmaking: Stereographic projection of Earth
  • 9 Proving: the essence of mathematics
  • 2 M.C. Escher: Stereographic projection of the Platonic solids
  • 3 Four-dimensional polyhedra: Simplex, hypercube, 24-cell, 120-cell, 600-cell
  • 4 The three-sphere and 4D polyhedra viewed stereographically
  • 5 Complex numbers: Now a sphere is a “complex projective line”
  • 6 Transformations with complex numbers: Some Möbius (linear fractional) transformations, some fractals (quadratic)]
  • 7-8 Hopf fibration of three-sphere

Each video starts very simply and gets harder, but the computer graphics are so incredibly stunning that you will stay for the pictures even after you lose the narration. They are somewhat cumulative. Don’t jump into the middle. Number 9 can be watched at any time. The authors recommend last or second.

[Posted & edited by Dr. Gene Chase. Reviewed in American Mathematical Monthly, vol. 120, no. 3, March 2013, pp. 288-290.]

Ten Questions about Flipping a Mathematics Classroom

Dr. Gene Chase, guest blogger.

“Flipping the classroom” is doing problems in groups during class time, while listening to lectures and reading books during homework time. See previous blog post. It is a sufficiently vague and controversial method of teaching that I have questions to ponder rather than answers to push. What do you think? I’m focusing on the mathematics classroom especially in the light of the popularity of Khan Academy’s mathematics modules and because this is a math blog.

1. Will students learn more because they are discussing content together in groups in class? Or will they have trouble staying on task because the teacher can’t attend to all groups at once? We have no control over the schedules of students outside of school to get them to interact about content outside of class. Do your students interact about mathematics outside of class now? (If you are reading this as a student, do you interact with other students about mathematics outside of class?)

2. Will “just-in-time” teaching of content when in the middle of solving a problem be more meaningful and more motivational than lectures that “front load” a student with lots of answers that don’t yet have questions? Or will “just-in-time” teaching encourage the kind of thinking that says the answers are only one problem-solving step away? In Japan, students expect to struggle with a problem; in the US students expect a problem to have a ready answer.

3. If the outside readings or videos are in smaller segments than a class lecture would be (say 5 minutes) will this make the material more digestible? Will modularizing lessons help students especially with attention deficit disorder, or will these modules instead promote more scattered attention to the content.

4. Reading mathematics textbooks is a special skill. Mathematics texts need to be read with a pencil and paper at the rate of a line a minute; in contrast, fiction can be read relaxing on the couch reading at a page a minute. So students are tempted to start a mathematics homework assignment without reading the text, and then go back to the text on a problem-by-problem basis to find a problem like the one that they are working on. Will a flipped classroom help students to engage with their textbooks more actively?

5. Could the flipped classroom be a fad because videos are “hotter” than books? In Marshall McCluhan’s terms books are a “cooler” medium than videos because books demand more effort on the part of the reader. Showing videos in class, if they take up the whole class period, are a waste of time. Do teachers do that because students don’t have access to the media outside of class? Because it’s easy? Print is more effective than video in delivering content unless the video has interactive features. For example, a study showed that news from the printed New York Times was remembered better than news from the on-line New York Times.

6. Problem Based Learning (PBL) worksheets for use in class are very time-consuming to develop because they need to address multiple levels of student preparation, and time-consuming to evaluate. Unless you are using modules for outside of class prepared by others like Khan Academy, preparing the content for use outside of class is time-consuming as well. Could you flip part of a class? Perhaps assign listening to a narrated Powerpoint about a single topic, a Powerpoint that you used with your lecture in a previous semester? (Thanks to Dr. Jennifer Fisler of Messiah College for this suggestion.) Mathematics is skill-based. Could you “flip” a single skill?

7. At the college level, students are supposed to spend two hours outside of class for every hour in class. How can two hours be flipped with one hour? Outside material would have to be lecture plus half of the homework —the homework not covered in class the previous day. So we’re back to the traditional model. Laboratory sciences already recognize that PBL requires twice as much time as lecture, and they recognize that students will finish labs at different rates. Could mathematics be taught as as laboratory science? In the experimental sciences, concepts are exact, but the lab part can be messy. (Thanks to Dr. Richard Schaeffer of Messiah College for that observation in this context.)

8. Flipping a small class is easier than flipping a large class. Students who are home-schooled typically experience a small flipped class. Thirty students using PBL in an hour only allow a teacher to give individual help at the average rate of two minutes per student. Should the students be grouped heterogeneously so quicker students can help slower students?

9. Do I lecture because it’s energizing for me, whereas helping students at their desks is draining? Can I remain non-threatened by questions to which I don’t know the answer if I lose the control of the class that lectures afford?

10. This will only work for college, since secondary school teachers don’t have the option to ask students to leave the class because they didn’t do their homework. Would a “ticket to ride” be an extrinsic incentive to prepare for class by doing the reading or watching the videos in advance of the class? A “ticket to ride” is a little one-question pre-quiz at the start of class that gives students the opportunity to earn the right to attend the class.